Editing Self-similarity (section)

==Examples==
[[Image:Feigenbaumzoom.gif|left|thumb|201px|Self-similarity in the [[Mandelbrot set]] shown by zooming in on the Feigenbaum point at (−1.401155189...,&nbsp;0)]]
[[Image:Fractal fern explained.png|thumb|right|300px|An image of the [[Barnsley fern]] which exhibits [[affine transformation|affine]] self-similarity]]

The [[Mandelbrot set]] is also self-similar around [[Misiurewicz point]]s.

Self-similarity has important consequences for the design of computer networks, as typical network traffic has self-similar properties. For example, in [[teletraffic engineering]], [[packet switched]] data traffic patterns seem to be statistically self-similar.<ref>{{cite journal|last1=Leland|first1=W.E.|last2=Taqqu|first2=M.S.|last3=Willinger|first3=W.|last4=Wilson|first4=D.V.|display-authors=2|title=On the self-similar nature of Ethernet traffic (extended version)|journal=IEEE/ACM Transactions on Networking|date=January 1995|volume=2|issue=1|pages=1–15|doi=10.1109/90.282603|s2cid=6011907|url=http://ccr.sigcomm.org/archive/1995/jan95/ccr-9501-leland.pdf}}</ref>  This property means that simple models using a [[Poisson distribution]] are inaccurate, and networks designed without taking self-similarity into account are likely to function in unexpected ways.

Similarly, [[stock market]] movements are described as displaying [[self-affinity]], i.e. they appear self-similar when transformed via an appropriate [[affine transformation]] for the level of detail being shown.<ref>{{cite magazine | url=https://www.scientificamerican.com/article/multifractals-explain-wall-street/ | title=How Fractals Can Explain What's Wrong with Wall Street | author=Benoit Mandelbrot | magazine=Scientific American|
date=February 1999| author-link=Benoit Mandelbrot}}</ref> [[Andrew Lo]]  describes stock market log return self-similarity in  [[econometrics]].<ref>Campbell, Lo and MacKinlay (1991)  "[[Econometrics]] of Financial Markets ", Princeton University Press! {{ISBN|978-0691043012}}</ref>

[[Finite subdivision rules]] are a powerful technique for building self-similar sets, including the [[Cantor set]] and the [[Sierpinski triangle]].

Some [[space filling curves]], such as the [[Peano curve]] and [[Moore curve]], also feature properties of self-similarity.<ref>{{Cite web |last=Salazar |first=Munera |last2=Eduardo |first2=Luis |date=July 1, 2016 |title=Self-Similarity of Space Filling Curves |url=https://repository.icesi.edu.co/items/5f9b8cea-4787-7785-e053-2cc003c84dc5 |url-status=live |archive-url=https://web.archive.org/web/20250313193207/https://repository.icesi.edu.co/items/5f9b8cea-4787-7785-e053-2cc003c84dc5 |archive-date=March 13, 2025 |access-date=March 13, 2025 |website=Universidad ICESI}}</ref>

[[File:RepeatedBarycentricSubdivision.png|thumb|A triangle subdivided repeatedly using [[barycentric subdivision]]. The complement of the large circles becomes a [[Sierpinski carpet]]]]

=== In cybernetics ===
The [[viable system model]] of [[Stafford Beer]] is an organizational model with an affine self-similar hierarchy, where a given viable system is one element of the System One of a viable system one recursive level higher up, and for whom the elements of its System One are viable systems one recursive level lower down.

=== In nature ===
[[File:Flickr - cyclonebill - Romanesco.jpg|thumb|right|200px|Close-up of a [[Romanesco broccoli]]]]
{{further|Patterns in nature}}
Self-similarity can be found in nature, as well. Plants, such as [[Romanesco broccoli]], exhibit strong self-similarity.

=== In music ===
* Strict [[canon (music)|canons]] display various types and amounts of self-similarity, as do sections of [[fugue (music)|fugues]].
* A [[Shepard tone]] is self-similar in the frequency or wavelength domains.
* The [[Denmark|Danish]] [[composer]] [[Per Nørgård]] has made use of a self-similar [[integer sequence]] named the 'infinity series' in much of his music.
* In the research field of [[music information retrieval]], self-similarity commonly refers to the fact that music often consists of parts that are repeated in time.<ref>{{cite book |last1=Foote |first1=Jonathan |title=Proceedings of the seventh ACM international conference on Multimedia (Part 1) |chapter=Visualizing music and audio using self-similarity |date=30 October 1999 |pages=77–80 |doi=10.1145/319463.319472 |url=http://musicweb.ucsd.edu/~sdubnov/CATbox/Reader/p77-foote.pdf |url-status=live |archive-url=https://web.archive.org/web/20170809032554/http://musicweb.ucsd.edu/~sdubnov/CATbox/Reader/p77-foote.pdf |archive-date=9 August 2017|isbn=978-1581131512 |citeseerx=10.1.1.223.194 |s2cid=3329298 }}</ref> In other words, music is self-similar under temporal translation, rather than (or in addition to) under scaling.<ref>{{cite book |last1=Pareyon |first1=Gabriel |title=On Musical Self-Similarity: Intersemiosis as Synecdoche and Analogy |date=April 2011 |publisher=International Semiotics Institute at Imatra; Semiotic Society of Finland |isbn=978-952-5431-32-2 |page=240 |url=https://tuhat.helsinki.fi/portal/files/15216101/Pareyon_Dissertation.pdf |access-date=30 July 2018 |archive-url=https://web.archive.org/web/20170208034152/https://tuhat.helsinki.fi/portal/files/15216101/Pareyon_Dissertation.pdf |archive-date=8 February 2017}} (Also see [https://books.google.com/books?id=xQIynayPqMQC&pg=PA240&lpg=PA240&focus=viewport&vq=%221/f+noise+substantially+as+a+temporal+phenomenon%22 Google Books])</ref>